\magnification = 2200 %\magstep3
%\vsize=1.05\vsize

\def\UseTimesRoman{
\font\cmr=Times
\font\TR=Times at 10pt
\font\TRXII=Times at 12pt
\font\TRXIV=Times at 14pt
\font\TRXX=Times at 20pt
\font\TRXXIV=Times at 24pt
\font\TI=TimesI at 10pt     %Times Italic
\font\TB=TimesB at 10pt     %Times Bold
\font\TBI=TimesBI at 10pt   %Times Bold Italic
\font\TBIviii=TimesBI at 8pt
\font\TBIv=TimesBI at 5pt
%\font\TO=TimesO at 10pt  %Times Oblique (Times Roman, slanted 22%
  %    with EdMetrics)
\font\TO=TimesI at 10pt  %Times Oblique (Times Roman, slanted 22%
    with EdMetrics)

\font\TIVIII=TimesI at 8pt
\font\TRVIII=Times at 8pt
\font\TIVI=TimesI at 6pt
\font\TRVI=Times at 6pt

	     \font\tenrmscld=Times at 10 pt
        \font\sevenrmscld=Times at 7 pt
        \font\fivermscld=Times at 5 pt

        \font\teniscld=cmmi10 at 10.3 pt
        \font\seveniscld=cmmi10 at 7.21 pt
        \font\fiveiscld=cmmi10 at 5.15 pt
        \font\tensyscld=cmsy10 at 10.3 pt
        \font\sevensyscld=cmsy10 at 7.21 pt
        \font\fivesyscld=cmsy10 at 5.15 pt
        \font\tenexscld=cmex10 at 10.3 pt
        \font\tenbfscld=cmbx10 at 10.3 pt
        \font\sevenbfscld=cmbx10 at 7.21 pt
        \font\fivebfscld=cmbx10 at 5.15 pt

\font\Courier = Courier
\font\Symbol = Symbol

\def\Omega{\hbox{{\Symbol W}}}

\textfont0=\tenrmscld \scriptfont0=\sevenrmscld\scriptscriptfont0=\fivermscld
\def\rm{\fam0\tenrmscld}
\textfont1=\teniscld \scriptfont1=\seveniscld \scriptscriptfont1=\fiveiscld
\def\mit{\fam1} \def\oldstyle{\fam1\teni}
\textfont2=\tensyscld \scriptfont2=\sevensyscld \scriptscriptfont2=\fivesyscld
\def\cal{\fam2}
\textfont3=\tenexscld \scriptfont3=\tenexscld \scriptscriptfont3=\tenexscld
\def\it{\TI}
\def\sl{\TO}
\def\bf{\TB}
\def\rm{\TR}
%\def\tt{\ttCourier}
\def\tt{\Courier}
\def\abstractfont{\TRVIII}
\def\footnotefont{\TRVIII}
\def\tinyfont{\TRvi}
\def\smalltitlefont{\TRXII}
\def\titlefont{\TRXIV}
\def\bigtitlefont{\TRXX}
\def\verybigtitlefont{\TRXXIV}
\textfont9=\TBI \scriptfont9=\TBIviii \scriptscriptfont9=\TBIv
\def\mbi{\fam9}
\rm
        }

%\UseTimesRoman

\def\BR{\Bbb R}             % Besondere Buchstaben
\def\BC{\Bbb C}
\def\BI{\Bbb I}
\def\BN{\Bbb N}
\def\BQ{\Bbb Q}
\def\BS{\Bbb S}
\def\BZ{\Bbb Z}
\def\Tilde{$_{\hbox{\cmrXX \~{}}}$}
\def\ST{\hbox{\eu T }}
\def\SRS{\hbox{\eu RS}}
\def\i{\hbox{{\bf i}}}

\font\sc=cmcsc10 at 10 pt   %% or: at 10 pt
\font\eu=eusb10 %at 10 pt
\font\small=cmr8 at 8 pt
\font\cmrX=cmbx10 scaled \magstep 1 %% 12 point CM
\font\cmrXX=cmbx12 scaled \magstep 1 %%

\hsize 7 true in
\vsize 9 true in
\hoffset = -0.20 true in
\voffset -0.25 true in
\parskip=3pt

\overfullrule = 0pt



\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex

\input BoxedEPS
\SetTexturesEPSFSpecial
\HideDisplacementBoxes

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue -10pt



\cl {\bf About  Epicycloids and Hypocycloids }
\lf
\cl { See also the ATOs for Spherical Cycloids  }
\lf
\cl{\sc  Definition and tangent construction}

\Lf
Hypocycloids are obtained if one circle of radius \break $r=hh$
rolls on the  inside  of another circle of radius $R=aa$.
The angular velocity of the rolling circle is $fr$ times
the angular velocity of the fixed circle (negative for
hypocycloids). $fr$ has to be an integer for the hypocycloid
to be closed. The formulas do not actually roll one circle
around another, they represent the curve as superposition
of two rotations:
$$ \eqalign{
&fr   := (R-r)/(-r);  \cr
&c.x := (R-r)\cos(t) + r \cos(fr * t);\cr
&c.y := (R-r)\sin(t) + r \sin(fr * t);
}$$
Double generation: If one changes the radius of the rolling
circle from $r$  to  $R-r$ then these formulas are preserved, 
except for the parametrization speed. To view this in
3DXM replace $hh$ by  $aa - hh$.
\Lf
 Epicycloids are obtained if one circle of radius \break $r = -hh$
rolls on the  outside  of another circle of radius $R=aa$.
The angular velocity of the rolling circle is $fr > 0$  times
the angular velocity of the fixed circle (again an integer
for closed epicycloids).
$$ \eqalign{
&fr   := (R+r)/r; \cr
&c.x := (R+r)\cos(t) - r \cos(fr * t); \cr
&c.y := (R+r)\sin(t) - r \sin(fr * t); \cr
}$$
These formulas agree with those of the hypocycloids
except for the sign of $r$. We view them in 3DXM by
using negative $hh$.
\Lf
We can also use a drawing stick of length $ii*r$. 
The default morph shows this: $0.5 < ii  < 1.5$.
\Lf
These more general ($ii<>1$) rolling curves were important 
for Greek astronomy because the planets orbit the sun 
(almost) on circles. Therefore, when one looks at other 
planets from earth, their orbits are (almost) such rolling
curves. It is no surprise that many of these curves have
individual names: Astroid, Cardioid, Limacon, Nephroid
are examples in 3DXM.
\Lf
Tangent construction. \lf
Rolling curves have a very simple tangent construction. The point of
the rolling circle which is in  contact with the base curve has velocity zero
-- just watch cars going by. This means that the connecting segment 
from  this point of contact of the wheel to the endpoint of the drawing 
stick is the  radius of the momentary rotation. The tangent of the curve 
which is drawn by  the drawing stick is therefore orthogonal to this 
momentary radius. \lf
The 3DXM-demo draws the rolling curve and shows its tangents.

\bye